1. Nucleon-Nucleon Forces
And symmetries
the simplest nuclei
LHC Particle CERN

2. Agenda Nature of NN forces from gross nuclear properties
Reality check: Properties of the deuteron demand sophistication (tensor & exchange forces)
Describe simple NN systems quantitatively, derive wave function, estimate depth V0 of central potential, revisit angular momentum coupling (isospin), Clebsch-Gordan coefficients
NN isospin multiplets and their stability
Excursion into field theory
NN Interactions
W. Udo Schröder, 2012

3. Properties of Nucleons
proton
neutron
Masses mp = MeV/c2
mn = MeV/c2
Charges ep = +|e| en = 0
Spins (mechanical) sp = ½ ħ sn = ½ ħ
Nucleons are Fermions
Each has 2 possible orientations in space: spin-up or spin-down mn, mp = ±½
Neutrons and protons are very similar, with similar interactions.
Consider them approximately as 2 different realizations of the nucleon: Isotopic (“iso”) spin-up or down tz(n) = -½, tz(p) = +½
Isospin T of multi-nucleon system is a hypothetical nuclear order parameter: Similar T  similar structure (configuration).
NN Interactions
W. Udo Schröder, 2012

4. Gross Characteristics of Nucleon-Nucleon Forces
Deductions from simple observations (not ab initio calculations):
VNN < 0: is attractive (stable assemblies of N’s exist)
Has short range, VNN ≠0 for r ~ 1 fm (10-15 m)
At distances r~ 1 fm (10-15 m) |VNN| ≫ |VCoul|
VNN =0 at r ~ Å (atoms don’t feel it)
VNN ≫ 0: at r < 1 fm  repulsive core 1: Volume (A) ≈ A∙DV (A=1) (nuclei do not collapse, approximately constant density). 2: BE (A) ≈ A∙DBE (A=1)  BE/A ≈ const.
Vnp≈ Vnn ≈ Vpp (safe for Vcoul)  charge independence
VNN depends on relative nucleonic spin orientation. I=0 nuclei are the most stable nuclei.
Also: Must be consistent with Fermion nature (exchange)
NN Interactions
VNN
attractive
repulsive
W. Udo Schröder, 2012

5. Simple Nuclei: The Deuteron
Radius , approx. spherical Mass mD= u < mn+mp Binding energy BD=2.225 MeV = Dmc2 Deuteron spin Magnetic moment mD= mN Electric quadrupole moment Q0=2.82mb « pRD2 =580mb (≈spherical nucleus, central V) ?
g Detector
Reactor n g
H2O
Expt: n+p  D + g (2.22 MeV)
Neutrons from reactor are captured by hydrogen in H2O coolant  emits mono energetic g-rays Eg = 2.22 MeV
NN Interactions
W. Udo Schröder, 2012

6. The Deuteron: Important Subtleties
Magnetic dipole moment mD = mN (No electric d) mn+mp=( )mN = mN  mD ≈ mn+mp but not exactly ! Quadrupole moment Q0=2.82 mb > 0 small but not zero! Effects of L ≠ 0 ?  orbital magnetic moment + Q0.
NN Interactions
Deuteron ground state not simple  2 components: |ygs|≈ 98% L=0 + 2% L=2 (excited D state).
L=1 does not work (wrong parity, spatial symmetry).
Consequences for the NN force: Cannot depend on NN distance r only, change in L implies existence of a torque, due to force involving the spins sn and sp.
 non-radial force not changing radial NN force ( NN tensor force)
W. Udo Schröder, 2012

7. Non-Central NN Forces z y f x Spin Dependence of Vdd
Nucleons are s=1/2 particles with magnetic dipole moments m.
Orientation of m slaved by spin
Classical electromagnet coupling of two dipoles via tensor interaction  angular correlation.
Classical elm dipole-dipole interaction:
mIħ f z x y
quantization axis
Spin Dependence of Vdd
NN Interactions
Such a tensor force exists also for nuclear interactions: Retain same angular correlation, different from expt.
W. Udo Schröder, 2012

8. Central & Non-Central NN Interactions
“Paris potentials”
Symmetry arguments for the form, strength parameters  experiment.
NN Interactions
For S=0 only central potential.
W. Udo Schröder, 2012

9. Spin-and Isospin-Exchange Interactions
Fermions are indistinguishable  wave function Y(q1,q2)=-Y(q2,q1) antisymmetric : exchange operators PM(S)PB(ℓ)PH(T)=-1, Different force components mediated by different mesons  experiment input (scattering)
Wigner forces
Majorana forces
Bartlett forces
Heisenberg forces
NN Interactions
W. Udo Schröder, 2012

10. Summary: Nucleon-Nucleon Interactionsp± p0
nucleon exchange terms
Main conclusions: Complicated structure of forces deduced from binding energy, quadrupole moment of deuteron and from NN scattering, relative strengths from fits to data:
Main NN component has central character ( ).
Spin dependent central force (from n-p scattering).
Spin-orbit NN forces depend on L-S coupling.
Non-central (tensor) force (Q0 of deuteron).
NN force is of meson-exchange type (n-p scattering).
NN forces do not depend on charge, nuclear forces are similar for nn, pp, and np (charge independence).
Fermion character of nucleons prohibits certain interactions.
p(n,n’)p
NN Interactions
Illustration of exchange character of NN forces:
Backward scattering (q > 900) of projectile in high-energy n-p scattering. Classical expectation: P(q>900)=0 !
T. Hamada & I.D. Johnston, Ann. Rev. Nucl. Sci 24, 151 (1974)
W. Udo Schröder, 2012

11. NN Energy Spectrum Mainly central NN forces: Rotational Invariance
3D Square Well
Oscill.
Woods Saxon
Square Well
Oscillator
Woods Saxon
Mainly central NN forces: Rotational Invariance
 decoupled radial, angular motion  ℓ conserved
Mean Field
W. Udo Schröder, 2011

12. Two-Nucleon Systems: Deuteron
E=-Bd=-2.25 MeV
R=2 fm
NN Interactions
From R≈1.5fm  V0 ≈ 50 MeV
General, for all nuclei !
W. Udo Schröder, 2012

13. NN Interactions
W. Udo Schröder, 2012

14. NN Interactions
W. Udo Schröder, 2012

15. NN Interactions
W. Udo Schröder, 2012

16. Clebsch-Gordan Coupling Coefficients
Vibrational Model
W. Udo Schröder, 2005

17. Clebsch-Gordan Coupling Coefficients
Vibrational Model
W. Udo Schröder, 2005

18. 2-Nucleon Isospin Multiplet
-2.2 MeV n p
Dineutron
Deuteron
Diproton
Bound unbound
Tz = Tz = Tz = +1
Deuteron is one of 3 possible 2-nucleon configurations. Only 1 configuration (D) has a bound state (S=1 triplet) Dineutron & diproton are unbound.
T = T = 0,1 T = 1
NN Interactions
Two-nucleon configurations: Spin S=1 triplet dineutron & diproton are Pauli forbidden, S=0 singlets are also unbound.
Singlet-S=0, T=1 symmetric “Isospin-Triplet” is unbound.
W. Udo Schröder, 2012

19. Illustration: Exchange Forcesp n
Indistinguishable isomeric states e
Chemical Bonding p
Non-relativistic analog: N p
Hybrid bound+ unbound states
Energy Levels
Bonding
Anti-Bonding E0 K
NN Interactions
Exchange of different mesons (quarks) leads to different interaction energies, interaction potential strengths and ranges.
W. Udo Schröder, 2012

20. Interactions, Fields, and Quanta
Illustration of the modern idea of interactions at a distance: Fields guide quanta like a quantal wave function. Quanta are absorbed (better: exchanged). Exchanged particles transfer energy and momentum, here to electrons moving in the antenna stem.
Photons are the quanta of the electromagnetic field.
NN Interactions
Receiver
Emitter
W. Udo Schröder, 2012

21. Interaction Fields
Interactions are mediated by exchange of field bosons (photons, mesons,..quark model)
elm
nuclear ,…
NN Interactions
weak
weak
W. Udo Schröder, 2012

22. Interaction Fields
Interactions are mediated by exchange of field bosons (photons, mesons,..quark model)
elm
nuclear ,…
NN Interactions
weak
weak
W. Udo Schröder, 2012

23. The Quark-Lepton Model of Matter
Explains the consistency of the known particles in all of their states. 3
families of quarks (3 “colors” each) and associated leptons.
All are spin-1/2 particles, quarks have non-integer charges
W. Udo Schröder: History NS
Mesons (q, q-bar)
q-bar:anti-quark
Nucleons (q,q,q)
September 01

24. Questions/Exercises
In relative terms, is the deuteron quadrupole moment large or small? Give a reason.
Does the fact that there is a bound state of the deuteron but none for the dineutron or diproton violate the concept of a charge independent nuclear forces? Justify rationally your judgment.
Given that the nucleonic spin operator is defined as , where is the set of Pauli matrices, calculate the expectation values for the NN tensor force in the singlet and triplet states.
Write down a formal expression for the deuteron angular-momentum/spin wave function, with its two components.
NN Interactions
W. Udo Schröder, 2012

25. W. Udo Schröder: History NS
The End
W. Udo Schröder: History NS
2012

26. W. Udo Schröder: History NS
September 01

27. Single-Particle Wave Functions
1D Schrödinger problem
Square Well
Oscillator
Woods Saxon
Independent of m  E ≠E(m) R
en V(r)
un0(r)
Mean Field
W. Udo Schröder, 2011

28. Interactions between Nucleons
Nuclei are strongly bound. The separation (dissociation) of a nucleon from most nuclei costs energy of about 8-10 MeV
The nucleus can only be stable, if there is a specifically nuclear, strongly attractive force acting between the nucleons.
Nuclear attraction over-powers Coulomb repulsion.
“Saturates” a small distances
Agreement with findings in particle physics.
Coulomb
U(r)/MeV
Nuclear potential energies of interactions for p-p, n-n, n-p are very similar:
Charge Independence of Nuclear Force
Nuclear
NN Interactions
Distance/fm
W. Udo Schröder, 2012